analysis of lightning electromagnetic fields at …...xv international conference on atmospheric...

XV International Conference on Atmospheric Electricity, 15-20 June 2014, Norman, Oklahoma, U.S.A.

1

Analysis of Lightning Electromagnetic Fields at Near and

Far Ranges

Yazhou Chen1,2*

, Hao Wang2, Vladimir A. Rakov

1

1. University of Florida, Gainesville, FL, USA

2. Shijiazhuang Mechanical Engineering College, Shijiazhuang, Hebei, China

ABSTRACT: The waveforms of lightning return stroke electromagnetic fields on ground are studied

using the transmission-line model. Approximate expressions to calculate lightning electromagnetic fields

at near and far ranges are presented. It is found that, the waveforms of lightning electric and magnetic

fields in the time domain at near and far distances can be expressed approximately as the channel-base

current waveform multiplied by a factor which is a function of return stroke speed v, the speed of light c,

and the horizontal distance r between the return-stroke channel and the observation point on ground. The

ranges at which the approximate expressions are valid are determined.

INTRODUCTION

Lightning is a common natural phenomenon, and it is estimated that some tens to 100 flashes,

including cloud and ground flashes, occur every second on Earth. Cloud-to-ground flashes are of primary

concern from the lightning protection point of view. The effects of lightning electromagnetic fields have

caused significant problems in recent years [Rakov and Uman 2003; Rakov and Rachidi 2009; Miyazaki

et al. 2010; Samaras et al. 2007; Thottappillil 2002]. It is important to investigate the characteristics of

lightning electromagnetic fields using both theory and measurements in order to simulate the lightning

electromagnetic fields in laboratory and to determinate the threat level to electric systems. Many works

have been done in this regard [Hoole and Balasuriya 1993; Lin and Uman 1979; Jerauld et al. 2008;

Barbosa et al. 2008; Lee et al. 2004; Hussein et al. 2008; Cooray et al. 2004; Uman et al. 2000; Jerauld et

al. 2009; Rakov and Uman 1998; Cooray 1993]. Nucci et al. [1990] identified four characteristic features

in the lightning electromagnetic fields at 1 to 200 km measured by Lin et al. [1979]. Rakov and Uman

[1998] defined four classes of lightning return stroke models which can be used to relate electromagnetic

fields to lightning current. Uman et al. [1975] demonstrated the approximate relationship between the

lightning electromagnetic field at far ranges and channel-base current and showed that the far fields

beyond a certain distance can be inferred from the channel-base current. Uman et al. [2002] found that the

current derivative waveforms, the magnetic flux density derivative waveform, and electric field intensity

derivative waveforms are essentially unipolar pulses that have similar wave shapes at 15 m from the

triggered lightning channel. Similar structure was observed by Jerauld et al. [2007] in dE/dt at 15 m and

30 m and dI/dt waveforms produced by an unusual rocket-triggered lightning stroke, which involved a

downward dart-stepped leader and a pronounced upward connecting leader. The above studies have

Contact information:Yazhou Chen, University of Florida, Ganiesville, Florida, US, Email: [email protected]


2

connected to some extent the waveforms of lightning electromagnetic field with that of the channel-base

current. In this paper, in order to theoretically establish the relationship between the lightning

electromagnetic field and the channel-base current in more detail, we will derive the approximate

analytical expressions for the lightning electromagnetic fields on earth surface at near and far ranges,

using the transmission line (TL) model and the dipole method. The ranges at which the approximate

expressions are valid are established by comparing their predictions with the electromagnetic fields

calculated using the exact expressions.

EXPRESSIONS FOR LIGHTNING ELECTROMAGNETIC FIELD AT GROUND LEVEL

The return-stroke channel can be regarded as a vertical antenna, as shown in Fig. 1. The transient

current propagates along the vertical conductor (the lightning channel) above a perfectly conducting plane

(the ground).

Fig. 1. The geometry used in calculating lightning return stroke electromagnetic fields.

According to the dipole method [Stratton 1941], the lightning electromagnetic fields due to an

upward-moving return stroke at an observation point P , ,r z (in cylindrical coordinate system) can be

described as follows:

5

0

4 2 3

1 3 ( ')( , / )d

4

3 ( ') ( ') ( , / )( , / ) d

r

r z zh tE i z R ch R

r z z r z z i z t R ci z t R c z

cR c R t

(1)

2 2

5

0

2 2 2

4 2 3

1 2( ')( , / )d

4

2( ') ( , / )( , / ) d

z

z z rh tE i z R ch R

z z r r i z t R ci z t R c z

cR c R t

(2)


3

3 2

1 ( , / )( , / ) d

4

r r i z t R chH i z t R c z

h R cR t

(3)

Where 2 2' -2 'R z r zz is the distance between the observation point and the element dz′ in the

channel at height z′, L is the length of the return-stroke channel, h+ and h- are the heights of the current

front as “seen” by the observer at time t in the channel and in its image, respectively, c is the speed of light,

i is the return-stroke current, Er is the radial electric field, Ez is the vertical electric field, HΦ is the

azimuthal magnetic field. In equations (1) and (2), the first term represents the electrostatic field, the

second term is the induction electric field, and the third term is the radiation electric field. In equation (3),

the first term represents the induction component of the magnetic field and the second term is the radiation

component of the magnetic field.

In this paper, we focus on the electromagnetic fields at ground level (i.e., the height of the

observation point z=0). Thus, h+=h-=h, Er=0.

According to the TL model [Uman and McLain 1969], the return stroke current is given by

( , ) ( ) (0, )i z t u t z v i t z v (4)

where u(ξ) is the Heaviside function equal to unity for ξ>0 and zero otherwise, and v is the lightning

current propagation speed along the lightning channel.

Substituting i(z′,t) given by (4) into (2) and (3), we can obtain each component of the electric and

magnetic fields, and the total electric and magnetic fields as follows [Rubinstein and Uman 1989]:

Electrostatic field:

2 2

1

3/2 2 22 2

0

2, 1 1 3

(electrostatic) tan2 2

z

h rthi z t h hrv vE

rv cr r h rh r

(5)

Induction electric field:

1

2 2

0

, 3(induction) tan

4z

i z t h hrE

rc r h r

(6)

Radiation electric field:

2

3/22 2 2

02 2 1/2

,(radiation)

12

( )

z

i z t rE

hc h r

v c h r

(7)

Induction magnetic field:

1/2

2 2

,(induction)

2

i z t hH

r h r

(8)

Radiation magnetic field:


4

1/2

2 2 2 2

,(radiation)

2

i z t rH

ch r h h r

v

(9)

Total electric field at ground level:

2 2

2

3/22 2 3/20 2 2 2

2 2

2, 1

( , )2 1

z

h rthi z t rv vE r t

rv hh rc h r

v c h r

(10)

Total magnetic field at ground level:

1/2 1/22 2 2 2 2 2

,( , )

2

i z t h rH r t

cr h r h r h h rv

(11)

Where

2 2 2

2

1

1

ct ct rh

, and /v c .

APPROXIMATE EXPRESSIONS FOR LIGHTNING ELECTROMAGNETIC FIELDS AND

THEIR DERIVATIVES AT NEAR RANGES

Approximate expressions for lightning electromagnetic fields at near ranges

If we let ', / ',f z t t i z t , then according to the TL model:

'/

'/ '/

00

', d ( ) 0, '/ d 0, '/ d

0, d 0, 0, '/ 0,0

'/ .

t t t

z v

t z v t z v

i z u z v i z v i z v

i s s f t f t z v f

F t z v

(12)

Thus, the current integration in the electrostatic field component of equation (2) is:

', / d / '/ .t

i z R c F t R c z v

(13)

Using the Taylor series expansion, / '/F t R c z v can be written as:

2/ '/ / ' / / / '/

/ 0, / / '/ .

F t R c z v F t r c F t r c t r c t R c z v c

F t r c i t r c R r c z v

(14)

Substituting (13) and (14) into the first term of equation (2), we can obtain the first-level

approximation for the electrostatic field component as:


5

2 2

500

2 2

500

3

0 0

1 2

3 2 3

1 2 3electrostatic ', / d d '

2

1 2 3/ '/ d '

2

/ 0, /

2 2

tan /1 3 1 2 1.

22

h t

z

h

R rE i z R c z

R

R rF t R c z v z

R

F t r c i t r ch

R h

h rrh h r

c r v R h rR h R h R h

(15)

The induction field component contains 1c . Substituting the Taylor series expansion of

( , / )i z t R c in the second term of equation (2), we can obtain the first-level approximation for the

induction electric field component as:

12 2

2400 0

0, / 0, / tan /2 3 3induction d

2 2 2 2

h

z

i t r c i t r c h rR r hE z

c R c r R h

(16)

The radiation electric field component at near ranges is very small and can be ingored. Thus, adding

equations (15) and (16), we obtain the first-level approximation expression for the total electric field at

near ranges as:

2

3 3 3

0 0

/ 0, / 1 2 1

2 2z

F t r c i t r ch rh rE

v R h rR h cR h R h

(17)

Similarly, we can obtain the first-level approximate expression for the magnetic field at near ranges

as:

30

1 1' ' induction 0, / d ' 0, /

2 2

h r hH H i t r c z i t r c

R r R h

(18)

As r << L at near ranges, R h gradually approaches h when the current propagates upward along

the channel. Neglecting the 1R term and 3R term compared with 1r term in equation (17), we can

obtain the second-level approximate expressions for the electromagnetic field at near ranges as:

0

10, /

2zE i t r c

vr (19)

1

0, /2

H i t r cr

(20)

According to equations (19) and (20), the lightning electric and magnetic field waveforms can be

expressed approximately as the channel-base current waveform with a factor which is a function of the

horizontal distance r. It is important to note that these equations are based on the TL model according to

which the current propagates along the channel at a constant speed and without either distortion or

attenuation.


6

Further, the time-derivatives of the lightning electromagnetic fields at near ranges can be obtained as:

0

d 0, /d ( , ) d ( , )d ( , ) 1

d d d 2 d

z zi t r cE r t E r tE r t

t t t vr t

(21)

d , d , d , d 0, /1

d d d 2 d

H r t H r t H r t i t r c

t t t r t

(22)

Evaluation of the approximate expressions for lightning electromagnetic fields at near ranges

In order to obtain the range of distances at which the approximate expressions are valid, the lightning

channel-base current with an 8/20μs waveform is used to calculate the lightning electromagnetic fields.

The length of the lightning channel is set to L=7.5km, and the return stroke speed is v=1.3×108m·s

-1.

Moreover, the pulse function [Zhang and Liu 2002] is chosen to describe the channel-base current,

expressed as follows:

01 2(0, ) 1 exp( ) exp( )

nIi t t t

(23)

where 1 2

2 1 2 1 1 2[ ( )] [ ( )]n

n n n

. In this paper, we set n=2. The other parameters of the

channel-base current are set as follows: 0I 30 kA, 5

1 4.0 10 s, 6

2 6.25 10 s. The electric and

magnetic fields calculated from the approximate expressions, including the first-level approximation

(equations (17) and (18)) and second-level approximation (equations (19) and (20)), along with the fields

calculated from exact electromagnetic field expressions (equations (10) and (11)) at different distances

from the lightning channel are shown in Fig. 2. Comparison between the electric and magnetic field

derivative waveforms from the approximate expressions, including the first-level and second-level

approximations, and the fields computed from exact expressions are given in Fig. 3.

As seen in Fig. 2, the waveforms predicted by the approximate expressions are similar to those based

on the exact expressions. The deviations of the approximate results from the exact results increase with

increasing distance r. The closer the distance, the less the deviation from the exact results. The

approximate electric field waveforms are essentially coincident with the exact ones within 100 m. When

the distance reaches 500 m, there is a significant difference between the approximate electric field

waveforms and the exact ones. It can be seen in Fig. 2 that the difference between the exact magnetic field

and its two approximations is smaller than that between the exact electric field and its two approximations

at same distance. The magnetic field waveforms based on the two approximate expressions can hardly be

distinguished from the exact results until the distance reaches about 500 m. It is important to note that the

calculated electric field waveforms do not show the characteristic flattening after 15 μs or so (seen in

measured waveforms), which is due to inadequacy of the TL model at later times [Rakov and Uman

1998].

As seen in Fig. 3, the deviations of the approximate electromagnetic fields derivative waveforms

from the exact derivative waveforms also increase with the increasing distance r. The closer the distance,

the less the deviation. The approximate electric field derivative waveforms are essentially coincident with

the exact derivative waveforms within 100 m. When the distance reaches 500 m, there is a significant

difference between the approximate and exact electric field derivative waveforms. From comparison of the

waveforms of electric and magnetic field derivatives in Fig. 3, it can be seen that the difference between


7

the exact magnetic field derivative and its two approximations is smaller than that between the exact

electric field derivative and its approximation at the same distance. The approximate magnetic field

derivative waveforms can hardly be distinguished from the exact ones until the distance reaches 500 m.

0 20 40 60 80

0

20

40

60

80 Ez

Ez'

Ez''

E (

kV

•m-1

)

t (s)

r=50 m(a1)

0 20 40 60 80

0

20

40

60

80

100 H

H'

H''

H (

A•m

-1)

t s)

r=50 m(b1)

0 20 40 60 80

0

10

20

30

40 Ez

Ez'

Ez''

E (

kV

•m-1

)

t (s)

r=100 m(a2)

0 20 40 60 80

0

10

20

30

40

50 H

H'

H''

H (

A•m

-1)

t (s)

r=100 m(b2)

0 20 40 60 80

0

5

10

15

20 Ez

Ez'

Ez''

E (

kV

•m-1

)

t (s)

r=200 m(a3)

0 20 40 60 80

0

5

10

15

20

25 H

H'

H''

H (

A•m

-1)

t s)

r=200 m(b3)

0 20 40 60 80

0

2

4

6

8

t s)

Ez

Ez'

Ez''

E (

kV

•m-1

)

r=500 m(a4)

0 20 40 60 80

0

2

4

6

8

10

t s)

H

H'

H''

H (

A•m

-1)

r=500 m(b4)

Fig. 2. (a) electric and (b) magnetic field waveforms predicted by the approximate expressions (the first-level and

second-level approximations are marked by single and double primes, respectively) and those based on the exact

expressions (unprimed field symbols) at different distances from the lightning channel.


8

0 20 40 60 80

-4

0

4

8

12

t (s)

dEz/dt

dEz'/dt

dEz''/dt

dE

/dt

(kV

•m-1

•s

-1)

r=50 m(a1)

0 20 40 60 80

-5

0

5

10

15 dH/dt

dH'/dt

dH''/dt

dH

/dt

(A•m

-1•

s-1

)

t (s)

r=50 m(b1)

0 20 40 60 80

-2

0

2

4

6

8

t (s)

dEz/dt

dEz'/dt

dEz''/dt

dE

/dt

(kV

•m-1

•s

-1)

r=100 m(a2)

0 20 40 60 80-4

-2

0

2

4

6

8

10

t (s)

dH/dt

dH'/dt

dH''/dt

dH

/dt

(A•m

-1•

s-1

)

r=100 m(b2)

0 20 40 60 80

-1

0

1

2

3

t (s)

dEz/dt

dEz'/dt

dEz''/dt

dE

/dt

(kV

•m-1

•s

-1)

r=200 m(a3)

0 20 40 60 80-2

-1

0

1

2

3

4

t (s)

dH/dt

dH'/dt

dH''/dt

dH

/dt

(A•m

-1•

s-1

)

r=200 m(b3)

0 20 40 60 80

-0.5

0.0

0.5

1.0

1.5

t (s)

dEz/dt

dEz'/dt

dEz''/dt

dE

/dt

(kV

•m-1

•s

-1)

r=500 m(a4)

0 20 40 60 80

-0.5

0.0

0.5

1.0

1.5

t (s)

dH/dt

dH'/dt

dH''/dt

dH

/dt

(A•m

-1•

s-1

)

r=500 m(b4)

Fig. 3. (a) electric and (b) magnetic fields derivative waveforms predicted by the approximate expressions (the first-

level and second-level approximations are marked by single and double primes, respectively) and those based on the

exact expressions (unprimed field derivative symbols) at different distances from the lightning channel.

APPROXIMATE EXPRESSIONS FOR LIGHTNING ELECTROMAGNETIC FIELDS AND

THEIR DERIVATIVES AT FAR RANGES

Approximate expressions for lightning electromagnetic fields at far ranges

At far ranges, R r and r >> L (L is the length of lightning channel), the radiation field is the


9

dominant component of the electromagnetic field produced by lightning return stroke, the electrostatic and

induction components can be neglected. We can simplify the expressions of electromagnetic fields as

follows [Uman et al. 1975]:

2 00

0, / '/1radiation d

2

L

z z

i t r c z vE E z

c r t

(24)

0

0, / '/1radiation d

2

L i t r c z vH H z

cr t

(25)

Since v is constant, we obtain:

0, / '/ 0, / '/

'

i t r c z v i t r c z vv

t z

(26)

Thus, equations (24) and (25) can be written as:

2 00

0, / '/radiation d

2 '

L

z

i t r c z vvE z

c r z

(27)

0

0, / '/radiation d

2 '

L i t r c z vvH z

cr z

(28)

After integration, we get:

2

0

radiation 0, / 0, / /2

z

vE i t r c i t r c L v

c r (29)

radiation 0, / 0, / /2

vH i t r c i t r c L v

cr

(30)

When 0 , (0, ) 0i . So, when / /t L v r c (the return-stroke front has not reached the

channel top yet), equations (29) and (30) can be expressed as:

2

0

, radiation 0, /2

z z

vE r t E i t r c

c r (31)

, radiation 0, /2

vH r t H i t r c

cr

(32)

From equations (31) and (32), the lightning electric and magnetic field waveforms can be expressed

approximately by the channel-base current waveform with factors (different from those for near ranges)

which are each a function of the return stroke speed v, the speed of light c, and the horizontal distance r, as

long as the return stroke current has not reached the top of the channel. The negative sign in equation (31)

means that the direction of the electric field is opposite to the direction of propagation of current wave

(positive charge moving up).

The time derivatives of equations (31) and (32) are as follows:

rad

2

0

d , d , d 0, /

d d 2 d

zE r t E r t i t r cv

t t c r t

(33)


10

rad

d , , d 0, /

d d 2 d

H r t dH r t i t r cv

t t cr t

(34)

Evaluation of the approximate expressions for lightning electromagnetic fields at far ranges

Equations (31) and (32) can be used for calculation of electromagnetic field only when the horizontal

distance r is much larger than the channel height L. However, the rise time to the peak of the lightning

electric and magnetic fields is only some microseconds or less. The field derivative corresponding to the

wavefront is much larger than that corresponding to the tail. Thus, the effect of the electromagnetic field

derivatives in the initial several microseconds or less will be dominant.

In order to validate the approximate expressions at far ranges, the lightning channel-base current with

an 8/20 μs waveform was adopted to calculate lightning electromagnetic fields. Accordingly, the

parameters of the channel-base current were set as follows: 0 30I kA, 5

1 4.0 10 s , 6

2 6.25 10 s.

The lightning channel length was set to L=7.5 km, and the return stroke speed to 81.3 10v m/s. The

waveforms of the radiation fields, the scaled channel-base current and the exact electromagnetic fields at

distances 5 km and beyond are given in Fig. 4. The waveforms of radiation field derivatives, the scaled

channel-base current derivatives and the exact electromagnetic fields derivatives at different distances are

shown in Fig. 5. Here, two different factors are introduced to be used with the scaled channel-base current.

For the electric field and its derivative, the multiplying factor for the current and its derivative is 1q ;

while for the magnetic field and its derivative, the multiplying factor for the current and its derivative is

2q . According to equations (31)-(34), the values of 1q and 2q are set as follows:

1 2

02

vq

c r ， 2

2

vq

cr

As follows from Fig. 4, the differences among the waveforms of the radiation fields, the scaled

channel-base currents, and the exact electromagnetic fields reduce when the distance r increases. This is

not surprising, since at 5 and 10 km significant contributions from electrostatic and induction field

components (not accounted for in the approximate expressions) are present. The smaller the distance, the

greater the differences. However, the rising portions of the radiation field and scaled channel-base current

waveforms are essentially coincident with those of the exact electromagnetic field waveforms. The

deviations between the radiation field waveforms and the scaled channel-base current waveforms are

relatively small. These deviations are mainly caused by the approximate substitution of r for R in

equations (31) and (32). As seen in Figs. 4 (a3) and (b3), at a distance of 50 km, the risetimes of the exact

electromagnetic field, radiation field, and scaled channel-base current are nearly the same. There is still

some difference among the waveforms of the exact electromagnetic fields, the radiation fields, and the

scaled channel-base currents, because the distance of 50 km is only about 7 times larger than the length of

the channel, which can not be viewed as r >> L (the far field approximation condition is not satisfied

perfectly). When the distance is 100 km, as seen in Figs. 4 (a4) and (b4), the waveforms of radiation field

are essentially coincident with the scaled channel-base current waveforms. There is only slight difference,

which occurs at late times, between the waveforms of the total electromagnetic fields and the radiation

fields. It is important to note that the waveforms at 50 and 100 km do not show the characteristic

zero-crossing (seen in measured waveforms), which is due to inadequacy of the TL model at later times


11

[Rakov and Uman 1998].

0 10 20 30 40 50 60 70 80-50

0

50

100

150

200

250

300 E

z

Erad

q1•i

E

(V•m

-1)

t (s)

r=5 km(a1)

0 20 40 60 80-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6 H

Hrad

q2•i

H (

A•m

-1)

t s)

r=5 km(b1)

0 10 20 30 40 50 60 70 80

0

20

40

60

80

100 E

z

Erad

q1•i

E (

V•m

-1)

t (s)

r=10 km(a2)

0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25 H

Hrad

q2•i

H

(A•m

-1)

t s)

r=10 km(b2)

0 10 20 30 40 50 60 70 80

0

4

8

12

16 Ez

Erad

q1•i

E (

V•m

-1)

t (s)

r=50 km(a3)

0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

0.05 H

Hrad

q2•i

H

(A•m

-1)

t s)

r=50 km(b3)

0 10 20 30 40 50 60 70 80

0

2

4

6

8 Ez

Erad

q1•i

E (

V•m

-1)

t (s)

r=100 km(a4)

0 20 40 60 80

0.000

0.005

0.010

0.015

0.020

0.025 H

Hrad

q2•i

H (

A•m

-1)

t s)

r=100 km(b4)

Fig. 4. (a) electric and (b) magnetic field waveforms predicted by the radiation field expressions and those based on

the exact expressions (including all field components) at different distances from the lightning channel. Also shown

in each plot is the corresponding scaled current (approximate far-field equation introduced here).


12

0 10 20 30 40 50 60 70 80

-10

0

10

20

30 dEz/dt

dErad

/dt

q1•di/dt

dE

/dt

(V•m

-1•

s-1

)

t (s)

r=5 km(a1)

0 20 40 60 80

-0.02

0.00

0.02

0.04

0.06

0.08 dH/dt

dHrad

/dt

q2•di/dt

dH

/dt

(A•m

-1•

s-1

)

t (s)

r=5 km(b1)

0 10 20 30 40 50 60 70 80

-4

0

4

8

12

16 dE

z/dt

dErad

/dt

q1•di/dt

dE

/dt

(V•m

-1•

s-1

)

t (s)

r=10 km(a2)

0 20 40 60 80

-0.01

0.00

0.01

0.02

0.03

0.04

dH/dt

dHrad

/dt

q2•di/dt

dH

/dt

(A•m

-1•

s-1

)t (s)

r=10 km(b2)

0 10 20 30 40 50 60 70 80

-1

0

1

2

3 dE

z/dt

dErad

/dt

q1•di/dt

dE

/dt

(V•m

-1•

s-1

)

t (s)

r=50 km(a3)

0 20 40 60 80

-0.0025

0.0000

0.0025

0.0050

0.0075 dH

/dt

dHrad

/dt

q2•di/dt

dH

/dt

(A•m

-1•

s-1

)

t (s)

r=50 km(b3)

0 10 20 30 40 50 60 70 80

-0.4

0.0

0.4

0.8

1.2

1.6 dE

z/dt

dErad

/dt

q1•di/dt

dE

/dt

(V•m

-1•

s-1

)

t (s)

r=100 km(a4)

0 20 40 60 80

-0.001

0.000

0.001

0.002

0.003

0.004 dH

/dt

dHrad

/dt

q2•di/dt

dH

/dt

(A•m

-1•

s-1

)

t (s)

r=100 km(b4)

Fig. 5. (a) electric and (b) magnetic field derivative waveforms predicted by the radiation field expressions and those

based on the exact expressions at different distances from the lightning channel. Also shown in each plot is the corre-

sponding scaled current (approximate far field equation introduced here).

As seen in Fig. 5, the deviations between the approximate results and exact result decrease with

increasing r. The closer the distance, the greater the deviations. The time derivative waveforms of the

radiation field and the scaled channel-base current are basically coincident with those of the total (exact)

electromagnetic field in the initial several microseconds at distances beyond 5 km. This is because the


13

height the return stroke current arrives at in the initial several microseconds is generally several hundreds

of meters. Thus, at distances beyond 5 km, the far field approximation condition is satisfied.

Comparing Fig. 4 with Fig. 5, we can see that the differences among the exact field derivatives, the

radiation field derivatives, and the scaled channel-base current derivatives are smaller than the differences

among the exact fields, the radiation fields, and the scaled channel-base currents at the same distance. It

can be concluded that the approximation condition for the electromagnetic fields at far ranges is more

strict than that for their derivatives. The approximate expressions for electromagnetic field derivatives are

valid at distances beyond several kilometers, and the approximate expressions for electromagnetic fields

are valid beyond 50 km.

SUMMARY AND CONCLUDING REMARKS

Based on the TL model, the approximate expressions for the lightning electromagnetic field on the

earth surface at near and far ranges are obtained. The approximate expressions show that the electric and

magnetic fields waveforms can be expressed approximately by the channel-base current waveform with a

factor which depends on distances. The validity of the approximate expressions is analyzed by comparing

the waveforms calculated from the approximate expressions at different ranges with the corresponding

exact expressions. The results show that the ranges at which the approximate expressions are valid at near

ranges are within 100 m, 100 m, 500 m, and 500 m for the electric field, electric field derivative, magnetic

field, and magnetic field derivative, respectively. The ranges at which the approximate expressions are

valid at far ranges are beyond 50 km, 5 km, 50 km, and 5 km for the electric field, electric field derivative,

magnetic field, and magnetic field derivative, respectively. If the return stroke speed is increased, the

ranges will be extended. In the limit, if the return stroke speed were equal to the light speed c, the

waveforms of electromagnetic fields will be same as the waveform of the lightning channel-base current at

any distance from the lightning channel, as shown theoretically for the TL model by Thottappillil et al.

[2001].

It should be pointed out that there is no hump in the computed electromagnetic fields at near ranges

although it is often seen in the measured fields. However, it does not matter when only the rise time and

the initial peak are needed.

ACKNOWLEDGMENTS

This research was supported by National Natural Science Foundation Project of China under Grant No.

51377171 and by the NIMBUS Program.

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